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## Compact composition operators on Hardy-Orlicz and Bergman-Orlicz spaces

(1)
1
Daniel Li

#### Abstract

It is known, from results of B. MacCluer and J. Shapiro (1986), that every composition operator which is compact on the Hardy space $H^p$, $1 \leq p < \infty$, is also compact on the Bergman space ${\mathfrak B}^p = L^p_a (\D)$. In this survey, after having described the above known results, we consider Hardy-Orlicz $H^\Psi$ and Bergman-Orlicz ${\mathfrak B}^\Psi$ spaces, characterize the compactness of their composition operators, and show that there exist Orlicz functions for which there are composition operators which are compact on $H^\Psi$ but not on ${\mathfrak B}^\Psi$.

#### Domains

Mathematics [math] Functional Analysis [math.FA]

### Dates and versions

hal-00530387 , version 1 (28-10-2010)
hal-00530387 , version 2 (21-03-2011)

### Identifiers

• HAL Id : hal-00530387 , version 2
• ARXIV :

### Cite

Daniel Li. Compact composition operators on Hardy-Orlicz and Bergman-Orlicz spaces. 2011. ⟨hal-00530387v2⟩

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